Ezra rebuilt the temple called the Second Temple or Herod Temple because largely rebuilt in 19 BC

The Second Temple is also known as the Herod Temple because in 19 BC largely rebuilt it. The original second temple was still a large undertaking and constructed by the returning exiles during Ezra's time under Zerubbabel's leadership. Cyrus largely furnished the temple and provided the large amount of money needed for this big project.
(Click picture for larger view).

Ezra: Rebuilding the Temple

The Book of Ezra starts where of 2 Chronicles finishes by completing the unfinished story of the exile. Seventy years had passed, and God fulfilled His promise by bringing the exiles back to the land.

The book of Ezra is confusing because it starts long before Ezra the scribe comes on the scene. Ezra doesn't appear until chapter 7. About 57 years pass between chapters 6 and 7.

Jeremiah 25.11 Whole land shall be a desolation and ... seventy years
Ezra's Time Chart - Post exile
We should note that the later period never officially could be called a kingdom but only a 'people.' There were at least three waves of return exiles. Zerubbabel, Ezra and Nehemiah all led groups back. Ezra and Nehemiah worked together for a period. Book of Ezra focuses on rebuilding of temple and then followed Nehemiah with the rebuilding of Jerusalem's wall.

Biblical Foundations for Freedom